Abstract [en]

Let Omega subset of R-N ( N greater than or equal to 2) be an unbounded domain, and L-m be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L-1-solutions to L-m (in Omega) approximate all L-1-solutions to L-m (in Omega), provided there exist real numbers R-j --> infinity, epsilon greater than or equal to 0, and it sequence {y(j)} such that B(y(j), epsilon) boolean AND Omega = circle divide and \A(y(j), R-j, R-N\Omega)\/R-j(N) > epsilon For All j, where \.\ means the volume and [GRAPHICS] for z is an element of R-N, R > 0 and D subset of R-N. For m = 2, we can replace the volume density by the capacity-density. It appears that the problem is related to this characterization of largest sets on which a nonzero polynomial solution to L-m may vanish, along with its (m-1)-derivarives. We also study a similar approximation problem for polyanalytic functions in C.